In non-Hermitian systems, there are line-gap insulators and point-gap
insulators[1]}, [2]} in general. In the point-gap insulator,
there is a gap in \(|E|\) . On the other hand, there are two types of line-gap
insulators. A real-line gap topological insulator has a gap in Re\(\left[E\right] \) ,
while an imaginary-line-gap topological insulator has a gap in Im\(\left[ E\right] \) .
We first consider the case \(\xi >0\) . For \(\left|\gamma \right|<\gamma _{1}\) , the system is a non-Hermitian line-gap
topological insulator along the Re\(\left[ E\right] \) . The systems is
metallic for \(\gamma _{1}\le \left|\gamma \right|\le \gamma _{0}\) .
For \(\left|\gamma \right|>\gamma _{0}\) , the system is a
non-Hermitian line-gap topological insulator along the Im\(\left[ E\right] \) .
If \(\xi <0\) , the system is a real-line-gap topological insulator for
\(\left|\gamma \right|<\gamma _{0}\) , it is a metal for \(\gamma _{0}\le \left|\gamma \right|\le \gamma _{1}\)  and it is an
imaginary-line-gap topological insulator for \(\left|\gamma \right|>\gamma _{1}\) . We show the topological phase diagram in Fig.REF (b).
It is consistent with the real and imaginary parts of the energy in the
\(\gamma \) -\(\xi \)  plane as shown in Fig.REF (a).
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